Generation of ray class fields modulo 2, 3, 4 or 6 by using the Weber function. (English) Zbl 1427.11119
Summary: Let \(K\) be an imaginary quadratic field with ring of integers \(\mathcal{O}_K\). Let \(E\) be an elliptic curve with complex multiplication by \(\mathcal{O}_K\), and let \(h_E\) be the Weber function on \(E\). Let \(N \in \{2,\,3,\,4,\,6\}\). We show that \(h_E\) alone when evaluated at a certain \(N\)-torsion point on \(E\) generates the ray class field of \(K\) modulo \(N\mathcal{O}_K\). This would be a partial answer to the question raised by Hasse and Ramachandra.
MSC:
| 11R37 | Class field theory |
| 11G15 | Complex multiplication and moduli of abelian varieties |
| 11G16 | Elliptic and modular units |
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